Mathematics for the interested outsider

Category Theory Seminar?

There are vanishingly few topics courses available this semester here. I don’t know if this is standard or not, but I think I might go buggy if I don’t find something to do other than calculus any my own work. You guys are great, yeah, but there’s only so much of the old give-and-take that makes mathematical discussions so exciting.

So I’m sounding out interest in an informal seminar on category theory here in the department. I suppose it’d be sort of like the Secret Russian Seminar, in that it’s not really on the books. I don’t know if I’m qualified to start up an official class or anything, given my rank.

Basically I’ll give a rough sketch of the basics of categories (not even as detailed as what I’ve done here), aiming at getting to applications to logic, semantics, physics, and (of course) topology as soon as possible. I may even try to make it accessible to good undergraduate students, if only to convert that many more people to the dark side benefits of categories.

Something similar was organized at my university this summer. Most of the best undergrads participated, and enthusiasm was initially great. However, it petered out because people were submerged in definitions and abstractness, without ever seeing an application to the topics they knew something about; so people went back to concentrate on their respective summer projects.

Todd, I didn’t know that they’d spent time here. You don’t happen to know when, do you?

Phils, exactly my thought. Here on my weblog I’ve got the time and space to make a long, measured approach. In this seminar, I only have so many weeks to work with, so I have to get into the applications as quickly as possible.

I am studying inner-product spaces. I have noticed that the inner product, norm and angle are defined without any reference to a basis. So, I would think that the inner-product is independent of basis. But, if you introduce a basis, the coordinates of the vectors in the space change with respect to the basis, and the value of the inner product will reflect this.

John, thank you for your help. I am going to have to think about your comment that choosing an inner product is equivalent to choosing a basis. But, it is good to know that inner product and norm are not basis independent.

The blurb on the back cover of Mac Lane’s Categories for the Working Mathematician (1st Edition) says the book was based on lectures at Chicago, Canberra, Bowdoin, and Tulane — I’d guess he lectured there in the late 1960’s?

Ross Street was an assistant professor at Tulane for a while, I think probably in the early 70’s (he got his PhD in 1967 or 1968, and was a post-doc at U. Illinois, I believe before he came to Tulane). I believe he was back in Sydney by 1974 at the latest. Around 1972, perhaps?

Todd, that’s right. There have been a number of category theorists passing through here, but not so much right now. The biggest categorist other than me is Mislove, the guy who’s sponsoring it as a VIGRE seminar partly for the sake of his student. They do work on “domain theory”, which plays into semantics. Clearly, I’ll be trying to get someone to talk about that sort of stuff as a branch of the seminar.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.